Tangents of two concentric ellipses

Question

Suppose a tangent to the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ cuts the ellipse $\frac{x^2}{c^2}+\frac{y^2}{d^2}=1$ at points $P,Q$, and that the tangents to the second ellipse at $P,Q$ are perpendicular. Determine $\frac{a^2}{c^2}+\frac{b^2}{d^2}.$

I tried as: I take a point on inner ellipse and from that wrote the equation of tangent and then tried to find the intersection points to other ellipse . But this became too complicated.

Can anyone please give me how to proceed?

Answer 1

Try to prove first, then exploit, the following lemma:

Given the ellipse $E_1$ with equation $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$, the locus of points $P$ such that the tangents to $E_1$ drawn through $P$ are perpendicular is a circle centered at the origin with radius $\sqrt{a^2+b^2}$. Additionally, if $P_1,P_2$ are the tangency points of the tangents from $P$, the envelope of the $P_1 P_2$ lines is an ellipse $E_2$ concentric and confocal with respect to $E$.